DECOMPOSITION OF SINGULAR MATRICES INTO IDEMPOTENTS ADEL ALAHMADI, S. K. JAIN, AND ANDRE LEROY Abstract. In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempo- tents and apply these factorizations for proving our main results. We generalize works due to Laffey ([12]) and Rao ([3]) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems (cf. [3], Theorems 5 and 7 and [4]). We also consider singular matrices over B´ezoutdomains as to when such a matrix is a product of idempotent matrices. 1. Introduction and definitions It was shown by Howie [10] that every mapping from a finite set X to itself with image of cardinality ≤ cardX − 1 is a product of idempotent mappings. Erd¨os[7] showed that every singular square matrix over a field can be expressed as a product of idempotent matrices and this was generalized by several authors to certain classes of rings, in particular, to division rings and euclidean domains [12]. Turning to singular elements let us mention two results: Rao [3] characterized, via continued fractions, singular matrices over a commutative PID that can be decomposed as a product of idempotent matrices and Hannah-O'Meara [9] showed, among other results, that for a right self-injective regular ring R, an element a is a product of idempotents if and only if Rr:ann(a) = l:ann(a)R= R(1 − a)R. The purpose of this paper is to provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and to apply these factorizations for proving our main results. Proposition 14 and Theorem 22 fill in the gaps in Rao's proof of a decomposition of singular matrices over principal ideal domains (cf. [3], Theorems 5 and 7), and simultaneously gener- alize these results. We show that over a local ring R (not necessarily commutative), if every 2 × 2 matrix A with r:ann(A) =6 0 is a product of idempotent matrices, then R must be a domain (Theorem 9). We prove the existence of a decomposi- tion into product of idempotents for any matrix A with l:ann(A) =6 0; over a local domain (not necessarily commutative) with Jacobson radical J(R) = gR such that n \nJ(R) = 0 (Theorem 10). Let R be a B´ezoutdomain such that every 2 × 2 singular matrix is a product of idempotent matrices. Theorem 22 shows that if every 2×2 invertible matrix over R is a product of elementary matrices and diagonal matrices with invertible diagonal entries then every n × n singular matrix is a product of idempotent matrices; Key words and phrases. Idempotent, Singular Matrix, Local Ring, Principal Right Ideal Do- main, B´ezoutDomain, Hermite Domain, Injective Module, Regular Ring, Unit Regular Ring, Stable Range 1. 1 2 ADEL ALAHMADI, S. K. JAIN, AND ANDRE LEROY The converse of Theorem 22 is true for commutative B´ezoutdomain, that is, if every n×n singular matrix over such a domain is a product of idempotent matrices then every 2 × 2 invertible matrix is a product of elementary matrices and diagonal matrices with invertible diagonal entries (Corollary 21). Finally, Theorem 24 studies the condition when each right singular element of the endomorphism ring of an injective module is a product of projections. This shows, in particular, that each linear transformation of a vector space, which is right singular, is a product of projections if and only if the vector space is finite-dimensional. Let us now give the main definitions and fix our terminology. All rings considered are nonzero rings with an identity element denoted by 1, and need not be commutative. A ring R is called a local ring if it has a unique maximal right ideal (equivalently, unique maximal left ideal). For example, power series ring F [[x]] over a field F and the localization Z(p) of the ring of integers Z are local rings. A ring R is called projective-free if each finitely generated projective right (equivalently, left) module is free of unique rank. Every local ring is projective-free. A ring R is a principal right (left) ideal ring if each right (left) ideal is principal. A right R-module M is called injective if every R-homomorphism from a right ideal of R to M can be extended to an R-homomorphism from R to M: Clearly, every vector space over a field is injective. A ring R is called right self-injective if it is injective as a right R-module. A ring R is called von Neumann regular if for each element a 2 R, there exists an element x 2 R such that axa = a. A ring R is called unit regular if for each element a 2 R, there exists an invertible element u such that aua = a. A ring R is called Dedekind finite if for all a; b 2 R, ab = 1 implies ba = 1: An n × n matrix is called elementary if it is of the form In + ceij, c 2 R with i =6 j. A ring R has a stable range 1 if for any a; b 2 R with aR + bR = R; there exists x 2 R such that a + bx 2 U(R), where U(R) is the set of invertible elements of R. A ring R is right (left) B´ezoutif any finitely generated right (left) ideal of R is principal. Hermite rings have been defined differently by different authors in the literature. Following Kaplansky, we call R to be a right (left) Hermite ring, if for any two elements a; b 2 R there exists a 2 × 2 invertible matrix P and an element d 2 R such that (a; b)P = (d; 0) (P (a; b)t = (d; 0)t). Lam ([14], section I, 4) calls this ring as K-Hermite ring. By a Hermite (B´ezout)ring we mean a ring which is both right and left Hermite (B´ezout). Amitsur showed that a ring R is a right (left) Hermite domain if and only if R is a right (left) B´ezoutdomain. Theorem 16 in this paper provides an alternative proof of Amitsur's theorem. A ring R is GE2 if any invertible 2×2 matrix is a product of elementary matrices and diagonal matrices with invertible diagonal entries. P 2 n n A right unimodular row is a row (a1; : : : ; an) R with the condition i=1 aiR = R. A right unimodular row is completable if it is a row (equivalently, the bottom row) of an invertible matrix. An element a in a ring R will be called right (left) singular if r:ann (a) =6 0 (l:ann (a) =6 0). An element is singular if it is both left and right singular. U(R) will denote the set of invertible elements of a ring R. Mn×m(R) stands for the set of n×m matrices over the ring R. The ring of n×n matrices over R will be denoted by Mn(R). The group of n × n invertible matrices over R is denoted by GLn(R). DECOMPOSITION OF SINGULAR MATRICES INTO IDEMPOTENTS 3 2. Preliminaries. We begin with an elementary lemma which works like our reference table for the proofs of our results. Note that one can obtain additional factorizations from a given factorization into idempotent matrices by taking conjugations. 2 Lemma 1.( (Table) of factorizations)( )( Let R)(be any ring) and let a; b; c R. Then a 0 1 a 0 0 1 0 (a) = ; ( 0 0 ) ( 0 0 )( 0 1 )( 1 0 ) 0 0 0 0 1 0 1 0 (a0) = , ( a 0 ) ( a 1 )( 0 0 )( 1 0 )( ) a ac 1 a 0 0 1 0 1 c (b) = , ( 0 0 ) ( 0 0 )( 0 1 )( 1 0 )( 0 0 ) a 0 1 0 1 1 0 0 1 0 (b0) = , ( ca 0 ) ( c 0 )( 0 0 ) 0 1 a 0 ac a 1 a 0 0 (c) = , ( 0 0 ) ( 0 0 )( c 1 ) ca 0 0 c 1 0 (c0) = , a 0 (0 1 ) a( 0 ) a b b(b−1a) b (d) with b 2 U(R), = can be factorized as in (c) and ( ) ( 0 0 ) 0 0 a 0 (ab−1)b 0 = can be factorized as in (c0), b 0 b( 0) ( ) a b a a(a−1b) (e) with a 2 U(R), = can be factorized as in (b) and ( ) ( 0 0 ) 0 0 a 0 a 0 = can be factorized as in (b0). b 0 (ba−1)a 0 In the next lemma, we consider factorizations of n × n matrices: Lemma 2. Let R be any ring and A 2 M2(R) be either (a) (an elementary) matrix, 0 1 (b) , 1 0 (c) (a diagonal) matrix,( ) a b a 0 (d) or ; a; b 2 R: 0 0 b 0 ( ) A 0 Then, for n ≥ 3, the n × n matrix is a product of idempotent matrices, 0 0 where zero blocs are of appropriate sizes. Proof. We will treat the case when n = 3. The general case is similar. (a) Let us, for instance,0 choose an1 elementary0 matrix1 0 A = I2 +1ae012. 1 ( ) 1 a 0 1 0 0 1 a 1 1 0 0 1 a A = . Then @0 1 0A = @0 1 0A @0 1 0A @0 1 0A. 0 1 − 0 1 0 0 01 0 0 0 01 00 0 1a 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 0 (b) @1 0 0A = @0 1 0A @ 1 0 −1A @0 1 0A.